By the way, Barrow’s argument is about the simplest proof of the (first part of the) fundamental theorem of calculus I have ever seen, using just elementary Euclidean geometry, and using Euclid’s definition of a tangent line, to a curve that is convex upwards at p, as a line meeting the curve at p but otherwise lying below it near p. Thus he has to show that if the original function is y = f(x), and the area function under graph(f) from 0 to x is A(x), then the line L through p = (x,A(x)) with slope f(x), lies below graph(A) both to the left and right of p. I.e. then the slope of graph(A) at p is f(x), so that f is the derivative of its area function A.
He is assuming that the function f is monotone, say increasing, near x. Then he wants to show that for h = delta(x) > 0 positive, then delta(A) = A(x+h) - A(x) is greater than delta(L), the rise in the line L from x to x+h, and that the decrease in A is less than that of L when h = delta(x)<0 is negative. But looking back at the graph of f, one sees that delta(L) = slope of L times h = f(x).h =the area of the rectangle with height f(x) and base h, while delta(A) is the area under that portion of graph(f) also with base h. Since f is increasing, the rectangle with height f(x) lies under the graph of f when h >0, and lies above it when h < 0. Hence the line L which meets graph(A) at p = (x,A(x)), rises more slowly than A as we move to the right, and drops more rapidly as we move to the left, i.e. the line L lies entirely below the curve graph(A) near p. QED.
In particular the argument follows from understanding this picture, where the fixed vertical solid line has height f(x).
Notice that continuity of f is not used here, reflecting the fact that Euclid’s definition of a tangent line does not imply uniqueness of the line. So this proof shows only that the line with slope f(x) is tangent to graph(A) at (x,f(x)) in Newton’s sense, provided the tangent line exists in his sense, i.e. provided it is the unique line satisfying Euclid’s definition.
Euclid of course is aware of the importance of uniqueness of the tangent line, but since he is treating only the case of a circle, he is able, in Prop.16, Book III, to prove uniqueness in that case as a corollary of his definition, showing no other line can be interposed between the circle and his tangent line.
In my opinion, Newton may well have used that property in Euclid's Prop. 16 as inspiration for his own definition. At least a careful translation of the fact that no other line can be interposed between the tangent and the circle does yield the modern statement that the tangent is a limit of secants, (for a convex curve). I.e. given any other line making any positive angle e>0 with the tangent line, the circle eventually gets between these two lines; hence (the circle being convex), given any e >0, all secants to points near enough to p, are closer than e to the tangent line.
